Tuesday, January 02, 2007

Friedmann Is In the Air Today


Thanks to Gebar and Kea for hints at putting equations on blogger.

Einstein in 1917 proposed a static universe, spherical with fixed radius. To prevent the sphere from collapsing due to its own gravitation, Einstein added a fudge factor, a cosmological constant. A spherical space would collapse UNLESS it were already expanding. An expanding universe would have been one of the greatest predictions ever, but Einstein rejected the idea.

Alexander Friedmann was born in St. Petersburg in 1888. His mother was a pianist, his father a ballet dancer and composer. While studying physics, he became interested in applications to meteorology. During the First World War, he volunteered to use his knowledge to aid the new tactic of aerial bombing. He flew as an observer on hazardous missions, becoming known to both sides. On days when Russian bombs were on-target, the Germans would mutter "Friedmann is in the air today."

After the war, despite the distraction of the revolution, Friedmann found solutions to Einstein's equations for an expanding Universe. His 1922 paper "On the Curvature of Space" is subtitled "to demonstrate the possibility of a world in which the curvature of space is independent of the three spatial coordinates but does depend on time." R is a multiple of t, R = ct. The universe needed no cosmological constant.

The Belgian priest Georges Lemaitre independently proposed an expanding universe as an explanation for redshifts. At a conference in 1927, he approached Einstein with the idea. The famous scientist could find no flaw with the math, but was uncomfortable with an expanding universe. Einstein had read the 1922 paper and wrongly thought Friedmann had made a mathematical error.

The Friedmann equations work both in units of mass density or energy density. First Friedmann equates kinetic and potential energies on the surface of an expanding sphere:
$\frac12 r^\prime$^2 = GM/r = $4 \pi G$r^2$\rho$(c^2)/3
($r^\prime$/r)^2 = $8 \pi G \rho$(c^2)/3
As we know, mass and energy are equivalent by E = m c^2.

Second we equate acceleration with gravitational attraction:
$r^\prime\prime$ = -GM/(r^2) = $-4 \pi G r \rho$(c^2)/3
$r^\prime\prime$/r = $-4 \pi G \rho$ (c^2)/3

Einstein gave us the field equation for gravitation:
$ R_{uv} - \frac12 g_{uv} R = \kappa T_{uv}$
The stress-energy tensor $T_uv$ (seen above) normally has units of energy density $\rho$(c^2).

We have seen that Einstein can be reduced to a form corresponding to Friedmann:
($r^\prime$/r)^2 - $\frac13 \kappa \rho$(c^2) = 0
($r^\prime\prime$/r) + $\frac16 \kappa \rho$(c^2) = 0

Removing, for a moment, the c^2 from Friedmann, we would have:
$8 \pi G \rho$/3 = $\frac13 \kappa \rho$ (c^2)
$4 \pi G \rho$/3 = $\frac16 \kappa \rho$ (c^2)

To normalise his left-hand with his right-hand sides, Einstein chose $ \kappa = 8 \pi G$/(c^2). This led to decades of misconception that Relativity requires a fixed c. Some "geometrized" unit systems give $ \kappa = 8 \pi G$/(c^4), which is too convoluted to describe. Since T_uv has units of energy density, we must use the same for Friedmann:
$8 \pi G \rho$(c^2)/3 = $\frac13 \kappa \rho$(c^2)
$4 \pi G \rho$(c^2)/3 = $\frac16 \kappa \rho$(c^2)
$\kappa = 8 \pi G$
This is very trivial. In earlier posts I've left out the (c^2) on both sides because it simply cancels out.

Now the Einstein equation is $ R_{uv} - \frac12 g_{uv} R = 8 \pi G T_{uv}$. The Bianchi identities become:
$\bigtriangledown {u} ( R_uv - \frac12 g_uv R ) = 0$
$8 \pi G \bigtriangledown {u} (T_uv) = 0$
The world is much simpler without that pesky (c^2) factor.

Finally, the Einstein-Hilbert action becomes:
$S = \int (16 \pi G R + L_m)$d^4(x)
Thus we can do everything General Relativity can without a fixed c. Some problems, like the deflection of bodies by the Sun, work even better with a varying c.

During the 1920's Edwin Hubble and Milton Humason used Cepheid Variables to show that the Universe expanded. In a 1931 visit to Mount Wilson, Einstein conferred with Hubble and peered through the telescope. Einstein dropped the cosmological constant, later calling it his "greatest blunder." He became one of Lemaitre's biggest supporters. If Einstein says the CC is a blunder it probably is a blunder.

Unfortunately, Friedmann had died in 1925 at the age of 37. He had continued his interests in weather and flying. After attempting a record-setting balloon flight, he contacted a fever. Inventing a Theory that makes real predictions (light is slowing down) runs the risk of being proved wrong. At first, even a great man like Einstein may think you have made an error. Whether challenging the air or contradicting Einstein, Alexander Friedmann was not afraid of taking risks.

9 Comments:

Blogger nige said...

Louise, please use html mark up tags sup and /sup (each inside < and > for superscript equations on blogger, and tags sub and /sub for subscript, while for mathematical symbols Pi and Kappa use

{<}FONT FACE="Symbol"{>}p{<}/FONT{>}

and

{<}FONT FACE="Symbol"{>}k{<}/FONT{>}

respectively, without the { and } curly brackets I had to add to post this comment.

One way to get the html mark up automatically done is to write the equations complete with superscript, subscript and inserted symbols in Word, then save as html. You can then open the Word file using Notepad, and copy the html code lines you need into blogger (set the blogger writer to show code when inserting html).

1:12 AM  
Blogger nige said...

I'm not a Friedmann hater. I just think he made a serious error in trying to describe cosmology using the field equation of general relativity.

There are numerous metrics, Friedmann had a whole landscape of solutions. You don't automatically get anything specific as a fixed prediction out of general relativity, it depends on the assumptions you put in about dark energy (cosmological constant value) and so on.

In addition, the whole Friedmann thing is based on gravity's effect on the expansion of the universe. Who has any evidence there is such a thing? How can you speculate about it without quantum gravity? One of the few things known for certain about quantum gravity is that if it is related to the Standard Model forces at all, they're all Yang-Mills quantum field theories.

Exchange radiation between charges or masses causes force. In the solar system or inside an atom or nucleus, there is no recession of the charges and masses to speak of, so redshift of gauge boson exchange radiation doesn't weaken the forces.

That's not true when masses are receding from one another rapidly.

From Yang-Mills quantum gravity arguments, with gravity strength depending on the energy of exchanged gravitons, the redshift of gravitons must stop gravitational retardation being effective.

This is not troublesome, it was published October 1996, and confirmed by observations two years later by Perlmutter. The trouble is not the above prediction, which is entirely successful. The trouble is that Perlmutter's result was compensated by the false invention of dark energy to cancel out long range gravity, instead of redshift

1:24 AM  
Blogger nige said...

The irony is that Friedmann's solutions cover a whole landscape of cosmologies, EXCLUDING the real one.

Even the lambda-CDM mainstream fiddle is not a good fit. You can't exactly cancel out long range gravity by adding a repulsive gravity (dark energy) which gets bigger with increasing distance; that overcompensates at extreme ranges. Hence the problem with the mainstream fiddle that the "dark energy" evolves: http://www.google.co.uk/search?hl=en&q=evolving+dark+energy&meta=

1:28 AM  
Blogger L. Riofrio said...

Thanks for the tip. I am experimenting with different ways of putting equations up, so funny things may appear from time to time.

Friedmann offered us multiple solutions. There is certainly room to improve upon old ideas, so I hope that your theory gets a hearing.

7:55 AM  
Blogger nige said...

Thanks Louise, but it's sad if an interlinking of facts determined by other people is personally mine just because I'm trying to make some noise to get the results heard.

1. Minkowski's spacetime 1907 says: time = distance/c

2. Hubble 1929 says recession velocity of mass is v = H.distance where H is constant

3. Putting Minkowski's spacetime into Hubble's equation gives v = Hct hence observable v (in spacetime we can observe, where light was emitted in the past) is increasing with time past we observe. Linear acceleration a = dv/dt = Hc ~ 6*10^{-10} ms^-2

4. Newton 1687 empirically suggests that mass accelerating implies outward force: F = ma. Putting in the Hubble acceleration from (3) a = dv/dt = Hc ~ 6*10^{-10} ms^-2, and the mass of surrounding (receding) universe m gives F ~ 10^{43} Newtons.

5. Newton 1687 empirically suggests that each action has an equal reaction force: F_action = -F_reaction. This predicts a reaction force of similar magnitude but opposite direction to the force in step (4) above.

6. The Standard Model physics empirically shows that the only stuff known below the IR cutoff (ie, which exists over vast distances) and which is also forceful enough to carry such a big inward force is Yang-Mills gauge boson exchange radiation.

7. The inward force predicts a universal attractive force of 6.7 x 10^{-11} mM/r^2 Newtons which is correct to two significant figures. I claim that nobody else can predict gravity on the basis of empirical facts; I've searched for a decade and nobody else can do this. There is no paper anywhere on the internet or in any journal predicting gravity. (Everyone else uses measured G instead of calculating it from other data and a causal mechanism for gravity based entirely on observed hard facts. Some people weirdly think Newton had a theory of gravity which predicted G, or that because Witten claimed in Physics Today magazine in 1996 that his stringy M-theory has the remarkable property of "predicting gravity", he can do it. The editor of Physical Review Letters seemed to suggest this to me when claiming falsely that the facts above leading to a prediction of gravity etc is an "alternative to currently accepted theories". Where is the theory in string? Where is the theory in M-"theory" which predicts G? It only predicts a spin-2 graviton mode for gravity, and the spin-2 graviton has never been observed. So I disagree with Dr Brown. This isn't an alternative to a currently accepted theory. It's tested and validated science, contrasted to currently accepted religious non-theory explaining an unobserved particle by using unobserved extra dimensional guesswork. I'm not saying string should be banned, but I don't agree that science should be so focussed on stringy guesswork that the hard facts are censored out in consequence!)

BTW, how is your book coming along?

5:48 AM  
Blogger nige said...

If you need bandwidth to host stuff on the internet, I can spare some at http://quantumfieldtheory.org which has a lot of space to spare. (I'm paid up for the domain name for 10 years and have paid two years hosting up front, and it is reasonably secure, with the servers in a disaster-proof building at Canary Wharf in London.)

5:55 AM  
Blogger L. Riofrio said...

I understand the frustration, but the good news is that stringers and dark energy types don't have a theory. If they pursue DE for 1500 years they still won't find anything. I would love to see your prediction of G written up somewhere, it sounds convincing.

The book is nearly done, though I will be tweaking bits of it for a while. Currently I am shopping it around, eagerly awaiting news from one particular publisher.

11:18 AM  
Blogger Kea said...

Wow. Good luck with the publishing! I can't wait to see it.

6:15 PM  
Anonymous Anonymous said...

You have the energy density, energy flux, momentum density and pressure correctly labeled but the section labeled viscosity should have been labeled shear. Although the link discusses stress in three dimensions, it is identical to the colored part of your stress graphic.

12:09 PM  

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